I know how to solve this, but why is the below reasoning wrong and leads to a mistake (I don't see any mistake!)
Step 1: From first equation $x=\dfrac{8}{y}$ , and $y$ is not zero
Step 2: From second equation $\dfrac{1}{x}=\dfrac{1}{4}+\dfrac{1}{y}\to\dfrac{1}{x}=\dfrac{y+1}{4y}\to x=\dfrac{4y}{y+1}$
Step 3: equate same $x$ from Step 1 and step 2: $\dfrac{8}{y}=\dfrac{4y}{y+1}$ , and $y$ is not $-1$ , but solving this for $y$ gives wrong result.
Where is the mistake in steps above?
The mistake is in the second step. In fact, it is $$\frac{1}{x} = \frac{1}{4}+\frac{1}{y} = \frac{y+4}{4y}$$